-                      r82592da
+                      r7e6bc45e
 ==================
 With two dimensional small angle diffraction data SasView will calculate
+With two dimensional small angle diffraction data sasmodels will calculate
 scattering from oriented particles, applicable for example to shear flow
 or orientation in a magnetic field.
 In general we first need to define the reference orientation
+of the particles with respect to the incoming neutron or X-ray beam. This
+is done using three angles: $\theta$ and $\phi$ define the orientation of
+the axis of the particle, angle $\Psi$ is defined as the orientation of
+the major axis of the particle cross section with respect to its starting
+position along the beam direction. The figures below are for an elliptical
+cross section cylinder, but may be applied analogously to other shapes of
+particle.
+of the particle's $a$-$b$-$c$ axes with respect to the incoming
+neutron or X-ray beam. This is done using three angles: $\theta$ and $\phi$
+define the orientation of the $c$-axis of the particle, and angle $\Psi$ is
+defined as the orientation of the major axis of the particle cross section
+with respect to its starting position along the beam direction (or
+equivalently, as rotation about the $c$ axis). There is an unavoidable
+ambiguity when $c$ is aligned with $z$ in that $\phi$ and $\Psi$ both
+serve to rotate the particle about $c$, but this symmetry is destroyed
+when $\theta$ is not a multiple of 180.
+The figures below are for an elliptical cross section cylinder, but may
+be applied analogously to other shapes of particle.
 .. note::
 …
     Definition of angles for oriented elliptical cylinder, where axis_ratio
     b/a is shown >1, Note that rotation $\theta$, initially in the $x$-$z$
+    b/a is shown >1. Note that rotation $\theta$, initially in the $x$-$z$
     plane, is carried out first, then rotation $\phi$ about the $z$-axis,
     finally rotation $\Psi$ is around the axis of the cylinder. The neutron
     or X-ray beam is along the $z$ axis.
+    or X-ray beam is along the $-z$ axis.
 .. figure::
 …
     with $\Psi$ = 0.
+Having established the mean direction of the particle we can then apply
+angular orientation distributions. This is done by a numerical integration
+over a range of angles in a similar way to particle size dispersity.
+In the current version of sasview the orientational dispersity is defined
+with respect to the axes of the particle.
+Having established the mean direction of the particle (the view) we can then
+apply angular orientation distributions (jitter). This is done by a numerical
+integration over a range of angles in a similar way to particle size
+dispersity. The orientation dispersity is defined with respect to the
+$a$-$b$-$c$ axes of the particle, with roll angle $\Psi$ about the $c$-axis,
+yaw angle $\theta$ about the $b$-axis and pitch angle $\phi$ about the
+$a$-axis.
+More formally, starting with axes $a$-$b$-$c$ of the particle aligned
+with axes $x$-$y$-$z$ of the laboratory frame, the orientation dispersity
+is applied first, using the
+`Tait-Bryan <https://en.wikipedia.org/wiki/Euler_angles#Conventions_2>`_
+$x$-$y'$-$z''$ convention with angles $\Delta\phi$-$\Delta\theta$-$\Delta\Psi$.
+The reference orientation then follows, using the
+`Euler angles <https://en.wikipedia.org/wiki/Euler_angles#Conventions>`_
+$z$-$y'$-$z''$ with angles $\phi$-$\theta$-$\Psi$.  This is implemented
+using rotation matrices as
+.. math::
+    R = R_z(\phi)\, R_y(\theta)\, R_z(\Psi)\,
+        R_x(\Delta\phi)\, R_y(\Delta\theta)\, R_z(\Delta\Psi)
+To transform detector $(q_x, q_y)$ values into $(q_a, q_b, q_c)$ for the
+shape in its canonical orientation, use
+.. math::
+    [q_a, q_b, q_c]^T = R^{-1} \, [q_x, q_y, 0]^T
+The inverse rotation is easily calculated by rotating the opposite directions
+in the reverse order, so
+.. math::
+    R^{-1} = R_z(-\Delta\Psi)\, R_y(-\Delta\theta)\, R_x(-\Delta\phi)\,
+             R_z(-\Psi)\, R_y(-\theta)\, R_z(-\phi)
 The $\theta$ and $\phi$ orientation parameters for the cylinder only appear
 when fitting 2d data. On introducing "Orientational Distribution" in
 the angles, "distribution of theta" and "distribution of phi" parameters will
+when fitting 2d data. On introducing "Orientation Distribution" in the
+angles, "distribution of theta" and "distribution of phi" parameters will
 appear. These are actually rotations about the axes $\delta_1$ and $\delta_2$
+of the cylinder, the $b$ and $a$ axes of the cylinder cross section. (When
+$\theta = \phi = 0$ these are parallel to the $Y$ and $X$ axes of the
+instrument.) The third orientation distribution, in $\Psi$, is about the $c$
+axis of the particle. Some experimentation may be required to understand the
+d patterns fully. A number of different shapes of distribution are
+available, as described for polydispersity, see :ref:`polydispersityhelp` .
+of the cylinder, which correspond to the $b$ and $a$ axes of the cylinder
+cross section. (When $\theta = \phi = 0$ these are parallel to the $Y$ and
+$X$ axes of the instrument.) The third orientation distribution, in $\Psi$,
+is about the $c$ axis of the particle. Some experimentation may be required
+to understand the 2d patterns fully. A number of different shapes of
+distribution are available, as described for size dispersity, see
+:ref:`polydispersityhelp`.
+Earlier versions of SasView had numerical integration issues in some
+circumstances when distributions passed through 90 degrees. The distributions
+in particle coordinates are more robust, but should still be approached with
+care for large ranges of angle.
+Given that the angular dispersion distribution is defined in cartesian space,
+over a cube defined by
+.. math::
+    [-\Delta \theta, \Delta \theta] \times
+    [-\Delta \phi, \Delta \phi] \times
+    [-\Delta \Psi, \Delta \Psi]
+but the orientation is defined over a sphere, we are left with a
+`map projection <https://en.wikipedia.org/wiki/List_of_map_projections>`_
+problem, with different tradeoffs depending on how values in $\Delta\theta$
+and $\Delta\phi$ are translated into latitude/longitude on the sphere.
+Sasmodels is using the *equirectangular* projection. In this projection,
+square patches in angular dispersity become wedge-shaped patches on the
+sphere. To correct for the changing point density, there is a scale factor of
+$\sin(\Delta\theta)$ that applies to each point in the integral. This is not
+enough, though. Consider a shape which is tumbling freely around the $b$
+axis, with $\Delta\theta$ uniform in $[-180, 180]$. At $\pm 90$, all points
+in $\Delta\phi$ map to the pole, so the jitter will have a
+distinct angular preference. If the spin axis is normal to the beam
+(which will be the case for $\theta=90$ and $\Psi=90$), the scattering
+pattern should be circularly symmetric, but it will go to zero at $\pm 90$ due
+to the $\sin(\Delta\theta)$ correction. This problem does not appear for a shape
+that is tumbling freely around the $a$ axis, with $\Delta\phi$ uniform in
+$[-180, 180]$, so swap the $a$ and $b$ axes so $\Delta\theta < \Delta\phi$
+and adjust $\Psi$ by 90. This works with the existing sasmodels shapes
+due to symmetry.
+There are alternative projections. The *sinusoidal* projection works by
+scaling $\Delta\phi$ as $\Delta\theta$ increases, and dropping those points
+outside $[-180, 180]$. The distortions are a little less for middle ranges of
+$\Delta\theta$, but they are still severe for large $\Delta\theta$ and the
+model is much harder to explain. The *Guyou* projection has an excellent
+balance with reasonable distortion in both $\Delta\theta$ and $\Delta\phi$,
+as well as preserving small patches. However, it is considerably more
+expensive to implement, and we have not yet computed the distortion
+correction, measuring the degree of stretch at the
+point $(\Delta\theta, \Delta\phi)$ in the correction.
 .. note::
+    Note that the form factors for oriented particles are also performing
+    numerical integrations over one or more variables, so care should be taken,
+    especially with very large particles or more extreme aspect ratios. In such
+    cases results may not be accurate, particularly at very high Q, unless the model
+    has been specifically coded to use limiting forms of the scattering equations.
+    For best numerical results keep the $\theta$ distribution narrower than the $\phi$
+    distribution. Thus for asymmetric particles, such as elliptical_cylinder, you may
+    need to reorder the sizes of the three axes to acheive the desired result.
+    This is due to the issues of mapping a rectangular distribution onto the
+    surface of a sphere.
+    Note that the form factors for oriented particles are performing
+    numerical integrations over one or more variables, so care should be
+    taken, especially with very large particles or more extreme aspect
+    ratios. In such cases results may not be accurate, particularly at very
+    high Q, unless the model has been specifically coded to use limiting
+    forms of the scattering equations.
+Users can experiment with the values of *Npts* and *Nsigs*, the number of steps
+used in the integration and the range spanned in number of standard deviations.
+The standard deviation is entered in units of degrees. For a "rectangular"
+distribution the full width should be $\pm \sqrt(3)$ ~ 1.73 standard deviations.
+The new "uniform" distribution avoids this by letting you directly specify the
+    For best numerical results keep the $\theta$ distribution narrower than
+    the $\phi$ distribution. Thus for asymmetric particles, such as
+    elliptical_cylinder, you may need to reorder the sizes of the three axes
+    to acheive the desired result. This is due to the issues of mapping a
+    rectanglar distribution onto the surface of a sphere.
+Users can experiment with the values of *Npts* and *Nsigs*, the number of steps
+used in the integration and the range spanned in number of standard deviations.
+The standard deviation is entered in units of degrees. For a "rectangular"
+distribution the full width should be $\pm \sqrt(3)$ ~ 1.73 standard deviations.
+The new "uniform" distribution avoids this by letting you directly specify the
 half width.
 The angular distributions will be truncated outside of the range -180 to +180
 degrees, so beware of using saying a broad Gaussian distribution with large value
+of *Nsigs*, as the array of *Npts* may be truncated to many fewer points than would
+give a good integration,as well as becoming rather meaningless. (At some point
+in the future the actual polydispersity arrays may be made available to the user
 for inspection.)
+The angular distributions may be truncated outside of the range -180 to +180
+degrees, so beware of using saying a broad Gaussian distribution with large
+value of *Nsigs*, as the array of *Npts* may be truncated to many fewer
+points than would give a good integration,as well as becoming rather
+meaningless. (At some point in the future the actual dispersion arrays may be
+made available to the user for inspection.)
 Some more detailed technical notes are provided in the developer section of
 this manual :ref:`orientation_developer` .
+This definition of orientation is new to SasView 4.2.  In earlier versions,
+the orientation distribution appeared as a distribution of view angles.
+This led to strange effects when $c$ was aligned with $z$, where changes
+to the $\phi$ angle served only to rotate the shape about $c$, rather than
+having a consistent interpretation as the pitch of the shape relative to
+the flow field defining the reference orientation.  Prior to SasView 4.1,
+the reference orientation was defined using a Tait-Bryan convention, making
+it difficult to control.  Now, rotation in $\theta$ modifies the spacings
+in the refraction pattern, and rotation in $\phi$ rotates it in the detector
+plane.
 *Document History*
 | 2017-11-06 Richard Heenan
+| 2017-12-20 Paul Kienzle

Note: See TracChangeset for help on using the changeset viewer.

SasView

Changeset 7e6bc45e in sasmodels for doc/guide/orientation/orientation.rst

Legend:

doc/guide/orientation/orientation.rst

Download in other formats: